This is an exposition of a general machinery developed by M. G. Eastwood, T. N. Bailey, C. R. Graham which analyses some real integral transforms using complex methods. The machinery deals with double fibrations $M\subset \Omega \stackrel{\eta }{\to }\leftarrow \tilde{\Omega }@>\tau >>X$ $(\Omega$ complex manifold; $M$ totally real, real-analytic submanifold; $\tilde{\Omega }$ real blow-up of $\Omega$ along $M$; $X$ smooth manifold; $\tau$ submersion with complex fibers of complex dimension one). The first result relates through an exact sequence the space of sections of a holomorphic vector bundle $V$ on $\Omega$, restricted to $M$, to its Dolbeault cohomology on $\Omega$, resp. its lift to $\tilde{\Omega }$. The second result proves a spectral sequence relating the involutive cohomology of the lift of $V$ to its push-down to $X$. The machinery is illustrated by its application to $X$-ray transform.

EUDML-ID : urn:eudml:doc:221095
Mots clés:

@article{701595,
     title = {Complex methods in real integral geometry},
     booktitle = {Proceedings of the 16th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1997},
     pages = {[55]-71},
     mrnumber = {MR1469021},
     zbl = {0902.53047},
     url = {http://dml.mathdoc.fr/item/701595}
}

Eastwood, Michael. Complex methods in real integral geometry, dans Proceedings of the 16th Winter School "Geometry and Physics", GDML_Books,  (1997), pp. [55]-71. http://gdmltest.u-ga.fr/item/701595/